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A higher order analogue of the Carathéodory–Julia theorem. (English) Zbl 1102.30028
The classical Carathéodory-Julia theorem states that if $$w$$ is in the Schur class $$S$$ of analytic mappings from the open unit ball $$D$$ into its closure and $$t_0\in T$$ is a boundary point of $$D$$, then the following are equivalent:
(1) $$d_1:= \liminf_{z \to t_0} \frac{1 -|w(z)|^2}{1 -|z|^2} < \infty.$$
(2) $$d_1:= \lim_{z {\widehat\to} t_0} \frac{1 -|w(z)|^2}{1 -|z|^2} < \infty.$$
(3) The limits $$w_0:= \lim_{z {\widehat\to} t_0} w(z)$$ and $$d_3:= \lim_{z {\widehat\to} t_0}\frac{1 - w(z) \overline w_0}{1 - z \bar t_0}$$ exist and satisfy $$|w_0| =1$$ and $$d_3 \geq 0.$$
(4) The limits $$w_0 := \lim_{z {\widehat\to} t_0} w(z)$$ and $$w_1 := \lim_{z {\widehat\to} t_0} w'(z)$$ exist and satisfy $$|w_0| =1$$ and $$t_0 w_1\overline w_0 \geq 0.$$
Here $$z {\widehat\to} t_0$$ means that $$z$$ approaches $$t_0$$ non-tangentially. Moreover, when these conditions hold, $$d_1 = d_2 = d_3 =t_0 w_1\overline w_0.$$
In this paper the authors prove a higher order analogue of the Carathéodory-Julia theorem. In order to formulate this result some notation must be introduced. The Schwarz-Pick matrix of the Schur function $$w \in S$$ is given by
${\mathbb P}^w_n(z):= \biggl(\frac{1}{i!j!}\frac{\partial^{i+j}}{\partial z^i \partial \overline z^j}\frac{ 1 -|w(z)|^2}{1 -|z|^2}\biggr)_{i,j=0}^n.$ For a given boundary point $$t_0\in T$$, the boundary Schwarz-Pick matrix is
${\mathbb P}^w_n(t_0)= \lim_{z \widehat\to t_0} {\mathbb P}^w_n(z), \;\;(n\geq 0),$ provided this limit exists. Further assume that $$w \in S$$ has non-tangential boundary limits $w_j(t_0):= \lim_{z \widehat\to t_0}\frac{w^{(j)}(z)}{j!}, \quad j=0,\dots,2n+1,$ and let $\mathbb P^w_n(t_0):= \begin{pmatrix} w_1(t_0) & \ldots & w_{n+1}(t_0) \\ \vdots & \;& \vdots \\ w_{n+1}(t_0) & \ldots & w_{2n+1}(t_0) \end{pmatrix} {\Psi}_n(t_0) \begin{pmatrix} \overline{w_0(t_0)} & \ldots & \overline {w_n(t_0)} \\ \;& \ddots & \vdots \\ 0 & \;& \overline{w_0(t_0)} \end{pmatrix},$ where the first factor is a Hankel matrix, the third factor is an upper triangular Toeplitz matrix and $${ \Psi}_n(t_0) = ( \Psi_{jl})_{j,l=0}^n$$ is the upper triangular matrix with entries $$\Psi_{jl} = 0$$, if $$j> l$$ and $$\Psi_{jl} = (-1)^l \binom{l}{j} t_0^{l+j+1}$$, if $$j \leq l$$. The lower right corner in the Schwarz-Pick matrix $${\mathbb P}^w_n(z)$$ is denoted by $d_{w,n}(z):=\frac{1}{(n!)^2}\frac{\partial^{2n}}{\partial z^n \partial \bar z^n}\frac{ 1 -|w(z)|^2}{1 -|z|^2}.$ The main result can be stated as follows: Theorem. For $$w \in S$$, $$t_0\in T$$ and $$n \in Z_+$$, the following are equivalent:
(1) $$\liminf_{z \to t_0} d_{w,n}(z) < \infty.$$
(2) $$\lim_{z {\widehat\to} t_0} d_{w,n}(z) < \infty.$$
(3) The boundary Schwarz-Pick matrix $${\mathbb P}^w_n(t_0)$$ exists.
(4) The non-tangential boundary limits $$w_j(t_0)$$ exists and satisfy $$|w_0(t_0)| =1$$ and $$\mathbb P^w_n(t_0)\geq 0,$$ where $$\mathbb P^w_n(t_0)$$ is the matrix defined above.
Moreover, when these conditions hold, the limits above in (1) and (2) are equal and furthermore, $${\mathbb P}^w_n(t_0)=\mathbb P^w_n(t_0).$$
This last equality makes it possible to compute boundary Schwarz-Pick matrices in terms of boundary values of $$w$$ and of its derivatives. This is in certain cases much easier to do than to use the original definition of $${\mathbb P}^w_n(t_0).$$ On the other hand, the conditions in (4) impose some restrictions on the boundary limits $$w_j(t_0)$$. When $$n=0$$ this main result reduces to the classical Carathéodory-Julia theorem with statement (3) excluded.

##### MSC:
 30D40 Cluster sets, prime ends, boundary behavior 46E20 Hilbert spaces of continuous, differentiable or analytic functions 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
##### Keywords:
Carathéodory-Julia theorem, boundary derivatives
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##### References:
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